Lemma 4.9.1. The linear operator l(c) in Def. 4.8.17 is non-expansive for any c∈E0. Proof. Fix c ∈ E0, set Ω := {ω > 0 : ω2 ∈ Sp(c c∗)}, ξ := idΩ and apply the represen-tations T : C0(Ω) → E0, L : C0(Ω) → L(E) used in (4.8.15) and (4.8.16). By (4.8.16) we have
l(c) = exp[−2L(log cosh(ξ))] =
= exp[−2b b∗] with
b :=T(√
log cosh(ξ)).
Hence the statement follows by axiom (P5).
According to Theorem 4.7.1, we have to see the existence of a bounded neighborhood B of 0 such that exp(Zc) is well defined on B for any c∈E0 and
sup
c∈E0,x∈B∥exp(Zc)(x)∥<∞. For this it suffices to establish that for some ϱ >0,
∑∞ k=0
1 k! sup
c∈E0
sup
∥x∥≤ϱ
dk0(zc)(x, . . . , x)<∞. (4.9.2) Notice that d00exp(Zc) = exp(Zc)(0) = e(c) lies in D0, the open unit ball of E0, indepen-dently of the choice of C ∈E0. Let us write
M := sup{∥{xc∗y}∥: x, y ∈E, c ∈E0, ∥x∥,∥y∥,∥c∥ ≤1}. We prove by induction onk that
∥Bk(e(c), x)∥ ≤Mk−1 (c∈E0,∥x∥ ≤1, k = 1,2, . . .). (4.9.3) Indeed, B1(e(c), x) = x (c∈E0, x∈E) and by the recursive Definition 4.8.17,
∥Bn+1(e(c), x)∥ ≤ 1 n
∑n ℓ=1
∥{Bℓ(e(c), x)e(c)∗Bn−ℓ+1(e(c), x)}∥ ≤
≤ 1 n
∑n ℓ=1
M ·Mℓ−1·Mn−ℓ =Mn
for all c ∈ E0, x ∈ E with ∥x∥ ≤ 1 whenever (4.9.3) holds for k = 1, . . . , n. In view of Proposition 4.8.18 and Lemma 4.9.1,
dk0exp(Zc)(x, . . . , x)≤k!Mk−1∥x∥k (c∈E0, x∈E)
for k = 1,2, . . .. Hence (4.9.2) is fulfilled for any ϱ < M−1. Corollary 4.9.4. In a partial JB∗-triple we have
exp(Zc)(x) = e(C) +ℓ(c) exp
({ze(c)∗z} ∂
∂z )
(x) (∥x∥ ≤M−1).
Proof. According to Def. 4.8.17 and (4.9.3), the expression F(t, e, x) := ∑∞
k=0
tkBk+1(e, x) satisfies
∂
∂tF(t, e, x) ={F(t, e, x)e∗F(t, e, x)} (∥e∥<1,∥x∥< M−1).
Therefore ∑∞
k=1Bk(e, x) is the power series of exp({ze∗z}∂/∂z) around 0 and this series converges uniformly on the domains {(e, x) : ∥e∥<1, ∥x∥ <(1−ε)M−1} for any ε > 0.
Thus the statement is immediate from Proposition 4.8.18.
Chapter 5
Extension of inner derivations from the base in partial J*-triples
Throughout this chapterE := (E, E0,{..∗.}) will stand for an arbitrarily fixed partial JB*-triple. Recall that, by axiom 4.1.4(P2), the operators ia a∗ (a∈E0) are derivations ofE.
Their finite real linear combinations are called inner derivations. In the sequel we shall call the spaceE0 the base of the partial Jordan-tripleEand we write E0 for the JB*-trple (E0,{..∗.} |E03). We shall also use the notation Der0(E) :={inner derivations of E}.
In 1990, Panou [71] axiomatized the J*-triples (
CK×CM,CK× {0},{..∗.})
admitting a bicircular domain D such that (x, y)∈D⇒(eiτx, eiϑy)∈D (τ, ϑ ∈R) in which all the vector fieldsZa = (a− {za∗z})∂/∂z (a∈CK× {0}) are complete. Actually he reobtained 4.1.4(P1-5) with hermitian operators with respect to a suitable Hilbert norm along with the property {E1E0∗E1} = 0 for E0 := CK × {0} and E1 := {0} ×CM. His main tool was the following fact [71] Prop.1.6: every inner derivation of the base of a partial Jordan-triple associated a finite dimensional bounded circular (not only bicircular) domain admits a unique inner derivative extension to the whole space.
As described in Chapter 4, in 1991, we gave [81] the complete algebraic axiomatics for the so-called geometric partial J*-triples which can be associated with some bounded circular domain in a Banach space. To obtain finer results concerning the structure of such partial Jordan-triples, it seems to be useful to look for infinite dimensional generalizations of [71] Prop.1.6. Notice that Panou’s proof relies heavily upon the fact that, in a finite di-mensional partial JB*-tripleE, the norm closureKof the group generated by exp(Der0(E)) is a compact subgroup of the unitary group with Lie-algebra Der0(E). In infinite dimen-sions, even if the base of the geometric partial Jordan-triple is finite dimensional, the group K may be non-compact.
In this chapter first we prove the analog of [71] Prop.1.6 to infinite dimensional partial J*-triples where the base is a finite dimensional Cartan factor. Our arguments are com-pletely Jordan-theoretical and require no further assumptions beyond the Jordan identity 1.10(J) and requiring only a a∗|E0 ∈ Her+(E0,∥.∥ |E0) (a ∈ E0). Then, by introducing the concept ofquasigridsand using norm-density considerations, we generalize the result to partial J*-triples whose base spaces arec0-direct sums of elementary JB∗-triples. In partic-ular we get that the restriction mapping ∆7→∆|E0 is a Lie-algebra isomorphism between Der0(E) and Der0(E0) whenever the base space E0 is a so called compact JB∗-triple.
5.1 The case of finite dimensional factor base
Lemma 5.1.1. Given a derivation ∆ of a partial J*-triple E and an element a ∈ E0 in the base E0 of E, we have
[∆, a a∗] = (∆a) a∗+a (∆a)∗. Proof. For any x∈E, by axiom (J2) we have
[∆, aa∗]x = ∆{a, a∗, x} − {a, a∗,(∆x)}={(∆a), a∗, x}+{a,(∆a)∗, x}
= ((∆a)a∗+a(∆a)∗)x.
Corollary 5.1.2. A derivation E ∈Der(E)vanishing on the baseE0 commutes with every inner derivation of E.
Proposition 5.1.3. Suppose the base E0 of the partial J*-triple E is a finite dimensional Cartan factor. Then for any inner derivation ∆0 of E0 there is a unique inner derivation
∆ of E with ∆0 = ∆|E0.
Proof. Let ∆0 ∈Der(E0). Using spectral decomposition [59], we can write
∆0 =i
∑n j=1
αjaj a∗j|E0
for some minimal tripotents a1, . . . , an ∈ E0 and α1, . . . , αn ∈ R. Now the operator i∑n
j=1αjaj a∗j is an inner derivation extending ∆0 to the whole E. Since real linear combinations of inner derivations are inner derivations, if suffices to show that
∑n j=1
αjaj a∗j = 0 whenever
∑n j=1
αj{ajaj∗c}= 0 (c∈E0)
and a0, . . . , an are minimal tripotents in E0. Since dim(E0) < ∞, the group of all linear isometries of E0 is a compact finite dimensional Lie group whose Lie algebra is Der(E0).
Consider its identity component U. That is, U is the closed subgroup of L(E0) generated by the operators exp(ia a∗) (a ∈E0). Then
Span(Ua) =E0 (0̸=a∈E0).
Indeed, the subspace Span(Ua) is Der(E0)-invariant. It is well-known [5] that Der-invariant subspaces are ideals in JB∗-triples, and the only ideals in the factor E0 are {0} and E0. By writing simply ∫
dU for the integration with respect to the normalized Haar measure1 onU,
∫
(U e) (U e)∗dU =
∫
(U f) (U f)∗dU (e, f minimal tripotents in E0)
1For Borel-measurable functions ϕ:U → L(E) with finite range,∫
ψdU :=∑
L∈ranψdU(ϕ−1(L))·L.
For continuous Φ :U → L(E), the Cauchy-type definition makes sense: ∫
ΦdU := limn
∫ϕndU whenever (ϕn)∞n=1 is a sequence of Borel functions of finite range tending uniformly to Φ.
because the group U is transitive (i.e. f =U e for some U ∈ U above) on the manifold of minimal tripotents in factors [59].
Let us fix α1, . . . , αn∈R along with minimal tripotents a1, . . . , an ∈E0 such that
∆c= 0, (c∈E0) where ∆ :=
∑n j=1
αjaj a∗j.
Since ∆|E0 = 0, by 5.1.2, ∆ commutes with any inner derivation ofE. Hence
∆ = exp ad(ia a∗)∆ = exp(ia a∗)∆ exp(ia a∗)−1 (a ∈E0),
∆ = U∆U−1 (U ∈ U),
∆ =
∫
U∆U−1dU =
∑n j=1
αjU(aj a∗j)U−1dU =
∑n j=1
αj(U aj) (U aj)∗dU
=
∑n j=1
αjZ where
Z :=
[ ∫
(U e) (U e)∗dU : e min.trip. ∈E0 ]
.
By Schur’s lemma, the operatorZ is a multiple of the identity when restricted toE0 (since Z|E0 commutes with U). Moreover
Z|E0 =γidE0 with some γ >0,
because e e∗ ̸= 0 is a positiveE-Hermitian operator whenever 0 ̸=e∈E0. Consequently, from the assumption ∆|E0 it follows ∑
jαj = 0 whence also ∆ =∑
jαjZ = 0.
Corollary 5.1.4. If the baseE0 of Eis a finite dimensional Cartan factor then the restric-tion map ∆ 7→ ∆|E0 of the complex operator Lie algebra D := SpanCE0 E0∗ is injective.
The center Z of D is 1-dimensional. The mapping P :∑
j
λjej e∗j 7→∑
j
λjZ (λj ∈C, ej min.trip.∈E0) is a well-defined contractive projection of D onto Z.
Proof. To prove the first statement, we only need to notice that any operator ∆∈ D has the form
∆ =A+iB, A=
∑n j=1
αjaj a∗j, B =
∑n j=1
βjbj b∗j
with α1, β1, . . . , αn, βn ∈ R. Since the operators A|E0, B|E0 are E0-Hermitian, ∆|E0
vanishes if and only if A|E0 =B|E0 = 0. By Proposition 5.1.3, the latter is equivalent to A=B = 0.
To see that the mapping P is a projection, observe that ifa∈E0 andU :=Ub|E0 where Ub := exp(iτ a a∗) and with τ ∈ R and a ∈E0 then (U e) (U e)∗ =Ub(e e∗)Ub−1 (e∈ E0).
Therefore every operator U ∈ U admits a (not necessarily unique) surjective isomorphic extension Ub from E0 to E such that (U e) (U e)∗ = Ub(e e∗)Ub−1 (e ∈E0). With such an extension operation
P∆ =
∫ Ub∆Ub−1dU (∆∈ D).
Since bU∆Ub−1=∥∆∥ ∥ for all U ∈U, the mapping P is contractive.
Remark 5.1.5. Although the closed subgroup U of L(E0) generated by the family {exp(ia a∗)|E0 : a ∈ E0} of operators is a finite dimensional compact Lie group if dim(E0) < ∞, the closed subgroup Ue of L(E) generated by {exp(ia a∗) : a ∈ E0} need not be a finite dimensional compact Lie group if the partial J*-triple E is infinite dimensional. The proof by Panou [71] using Levi-Malcev decomposition for the special case dim(E) <∞ of the proposition relies heavily upon the compactness of the group of surjective linear isometries of E.